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Omni-MATH / Let S={1,2,4,8,16,32,64,128,256}. A subset P of S is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set Q is called super squarely if…

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Let S={1,2,4,8,16,32,64,128,256}S=\{1,2,4,8,16,32,64,128,256\}. A subset PP of SS is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set QQ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets.
Plain-text mathematical notation (without MathML)
Let S={1,2,4,8,16,32,64,128,256}. A subset P of S is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set Q is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets.
Original LaTeX notation
Let $S=\{1,2,4,8,16,32,64,128,256\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets.

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